Practicing Success
Let f(x) be a continuous function such that f(a - x) + f(x) = 0 for all x ∈ [0, a]. Then $\int_0^a\frac{dx}{1+e^{f(x)}}$ is equal to: |
a $\frac{a}{2}$ f(a) $\frac{1}{2}f(a)$ |
$\frac{a}{2}$ |
f(x) = – f(a – x) $I=\int\limits_0^a\frac{dx}{1+e^{f(x)}}dx=\int\limits_0^a\frac{dx}{1+e^{f(a-x)}}=\int\limits_0^a\frac{dx}{1+e^{-f(x)}}=\int\limits_0^a\frac{e^{f(x)}}{1+e^{f(x)}}dx$ $⇒I=\frac{1}{2}\int\limits_0^a\frac{1+e^{f(x)}}{1+e^{f(x)}}dx=\frac{1}{2}(a-0)=\frac{a}{2}$ |